Universal potential barrier penetration by initially confined wavepackets

The dynamics of an initially sharp-boundary wavepacket in the presence of an arbitrary potential barrier are investigated. It is shown that the penetration through the barrier is universal in the sense that it depends only on the values of the wavefunction and its derivatives at the boundary. The dependence on the derivatives vanishes at long distances from the barrier, where the dynamics are governed solely by the initial value of the wavefunction at the boundary.